Method for Fitting Primitive Shapes to 3D Point Clouds Using Distance Fields

ABSTRACT

A method fits primitive shapes to a set of three-dimensional (3D) points by first converting the set of 3D points to a distance field. Each element in the distance field is associated with a distance to a nearest point in the set of 3D points. A set of two or more candidates are hypothesizing from the primitive shapes, and a score is determined for each candidate using the distance field. Then, the primitive shape to fit to the 3D points is selected from the candidates according to their scores.

FIELD OF THE INVENTION

This invention relates generally processing three-dimensional (3D) data, and more particularly to fitting primitive shapes to 3D data.

BACKGROUND OF THE INVENTION

Three-dimensional (3D) sensors based on structured light, laser scanning, or time of flight are used in many applications in robotics, computer vision, and computer graphics. The sensors scan a 3D scene as a set of 3D points, commonly referred to as a 3D point cloud. 3D point clouds for large scale scenes can be obtained by registering several scans acquired by the 3D sensors into a single coordinate system.

Storing and processing 3D point clouds require substantial memories and computational resources because each 3D point has to be stored and processed separately. Representing 3D point clouds as a set of primitive shapes is desired for compact modeling and fast processing.

One method fits primitive shapes to 3D point clouds using a random sample consensus (RANSAC) framework. That method hypothesizes several primitive, shapes and selects the best primitive shape according to scores of the hypothesized primitive shapes. That method uses raw 3D point clouds for determining the scores, which requires, for each hypothesized primitive shape, traversing all the points in the 3D point cloud or searching nearby points in the 3D point cloud given a reference point on the hypothesized primitive shape. This is time consuming.

SUMMARY OF THE INVENTION

The embodiments of the invention provide a method for fitting primitive shapes to 3D point clouds. The method represents the 3D point cloud by a distance field. A RANSAC framework efficiently performs the primitive shape fitting.

The distance field represents the distance from each point in a 3D space to a nearest object surface, which makes the score evaluation in the RANSAC framework efficient. The distance field also provides gradient directions for each point to the nearest object surface, which helps a refinement process based on gradient decent.

Thus, in contrast to the prior art, the use of the distance field representation allows for fast score computation for generated hypotheses in the RANSAC framework, and efficient refinement of hypothesized shape parameters.

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 is a flow diagram of a method for fitting primitive shapes to 3D point clouds using distance fields according to the embodiments of the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

As shown in FIG. 1, the embodiments of the invention provide a method for fitting primitive shapes to 3D point clouds using distance fields. Input to the method is a 3D point cloud 101. The 3D point cloud can be obtained as a scan of a 3D sensor, or by registering multiple scans of the 3D sensor or scans from several different 3D sensors.

The 3D point cloud is converted 105 to a distance field 102. The distance field is used in a RANSAC-based primitive shape fitting process 110, where a set of two or more candidate shapes 100 are hypothesized 111 by using a minimal number of points required to determine parameters of a corresponding shape. A score 121 is determined for each shape candidate.

The method selects 120 the best candidate primitive shape that has a minimal score among the candidates. Optionally, the parameters of the best primitive shape can be refined 130 using a gradient-decent procedure. To determine a set of primitive shapes in a point cloud, we iterate the process 110 after subtracting 140 the selected primitive shape from the distance field. The output of the method is a set of parameters 109 that define the set of primitive shapes.

The method can be performed in a processor 150 connected to memory and input/output interfaces as known in the art.

Conversion from Point Cloud to Distance Field

Let P={p_(i)}, for i=1, . . . , N, be a set of points of the 3D point cloud. We convert the point cloud to a distance field D(x), where x represents an element in the distance field. The element can be a 3D voxel in a regular cuboid grid having the volume V=W×H×L, where W, H, and L are width, height and length of the grid. Associated with an element or voxel is a distance to a nearest point in the set of 3D points.

Let R be a 3×3 rotation matrix, t be a 3×1 translation vector, and s be a scale factor to transform the coordinate system of the point cloud to the coordinate system of the distance field. Then, the i^(th) 3D point p_(i) in the point cloud is transformed to the coordinate system of the distance field as q_(i)=s(Rp_(i)+t). We determine R, t, and s such that all q_(i) are within the distance field of W×H×L voxels.

The transformed points q_(i) are discretized to the nearest grid point of the distance field as q_(i) ^(d)=round(q_(i)), where the function round(·) determines the nearest grid point of the 3D point in the argument. Let Q={q_(i) ^(d)} be the set of discretized points. The distance field of the point cloud is determined by solving the following minimization problem:

$\begin{matrix} {{{D(x)} = {\min\limits_{y \in Q}\left( {{d\left( {x,y} \right)} + {T(y)}} \right)}},} & (1) \end{matrix}$

where the function d determines a distance between the points x and y, and T is an indicator function

$\begin{matrix} {{T(y)} = \left\{ \begin{matrix} 0 & {{{if}\mspace{14mu} y} \in Q} \\ \infty & {{otherwise}.} \end{matrix} \right.} & (2) \end{matrix}$

A naïve solution to the minimization problem of Equation (1) requires O(V²) time. Procedures to determine the distance field in O(V) time are known for several distance functions d, such as Manhattan, Euclidean, and L₁ distances. Such procedures perform one-dimensional distance field computation sequentially for each dimension. Our preferred embodiment uses the Euclidean distance as the distance function.

After determining the distance field by solving Equation (1), we truncate the distance field as

$\begin{matrix} {{D(x)} = \left\{ \begin{matrix} {D(x)} & {{{if}\mspace{14mu} {D(x)}} < C} \\ C & {{otherwise},} \end{matrix} \right.} & (3) \end{matrix}$

where C is a threshold used for the truncation. This make the primitive fitting process accurate even when there are missing data.

Then, we determine a gradient vector field of D(x) which can be used to hypothesize the normal vector of a 3D point, as well as to determine the Jacobian matrix for the gradient-decent-based refinement process. We use the following equations to determine the gradient vector field:

$\begin{matrix} {{{\nabla{D(x)}} = \begin{pmatrix} {{D(x)} - {D\left( {x - \left( {1,0,0} \right)^{T}} \right)}} \\ {{D(x)} - {D\left( {x - \left( {0,1,0} \right)^{T}} \right)}} \\ {{D(x)} - {D\left( {x\left( {0,0,1} \right)}^{T} \right)}} \end{pmatrix}},{and}} & (4) \\ {{\overset{\_}{\nabla D}(x)} = {\frac{{\nabla D}(x)}{{{\nabla D}(x)}}.}} & (5) \end{matrix}$

To efficiently hypothesize primitive shapes, we store a set of voxels Z having zero distance, i.e., Z={x_(i)|D(x_(i))=0}. The voxels correspond to points on object surfaces, which we use to efficiently hypothesize primitive shapes.

The above preferred embodiment uses uniform-size elements or voxels. Note that adaptive-size voxels, such as octree-based representation can also be used.

Primitive Shape Fitting Using Distance Field

Given the distance field and the gradient vector field, we use the RANSAC framework for primitive shape fitting. We hypothesize a set of two or more candidate primitive shapes, determine their scores, and select the best candidate with the minimal score. In the preferred embodiments, we use infinite planes segments, spheres, cylinders, and cuboids as the primitive shapes. However, the number of different primitive shapes can be unbounded.

For each primitive, we use a minimum number of voxels required to generate a shape hypothesis. We select such voxels from the set Z. Note that each voxel has a position r_(i)∈Z and a normal n_(i)= ∇D(r_(i)) Thus, each voxel can be considered an oriented 3D point, i.e., 3D point having a 3D direction vector. Below, we describe the procedure for generating shape hypotheses for each primitive shape in the set 100.

Infinite Plane Primitives

An infinite plane, i.e., a plane without boundary, can be hypothesized by sampling a single oriented point (r₁,n₁) All points r on this plane should satisfy the following plane equation:

n ₁ ^(T)(r−r ₁)=0  (6)

To determine the score v for this plane, we sample several points r_(j) (j=1, . . . , J) within the volume of the distance field that satisfy

|n ₁ ^(T)(r−r ₁)|<ε,  (7)

where ε is a small threshold value, which allows small deviations of sampled points from the plane equation (6). We then average the distances of the sampled points as

$\begin{matrix} {v = {\frac{1}{J}{\sum\limits_{j = 1}^{J}{{D\left( r_{j} \right)}.}}}} & (8) \end{matrix}$

Hereafter, we denote a small threshold value as ε, which allows small deviations from exact values due to noise and discretization such as in Equation (7). Note that ε can be different for different equations.

Plane Segment Primitives

A plane segment primitive is represented by its plane equation and a set of points that are the members of the plane segment. First, we determine the plane equation by sampling a single oriented point (r₁,n₁) as in the infinite plane primitives. Then, we define a set of points on this plane connected from the sampled point position r₁. Specifically, we find neighboring points r_(m) from r₁ that satisfy Equation (7) and D(r_(m))<ε. We maintain a set of points r_(m) (m=1, . . . , M) as the member of the plane segment hypothesis. To determine the score for the hypothesis, we sample several points r_(j) from r_(m) and average the distances of the sampled points as in Equation (8).

Sphere Primitives

A sphere can be hypothesized by sampling two oriented points (r₁,n₁) and (r₂,n₂). For the two points to lie on a sphere, the line passing through r₁ and having the direction n₁ should intersect with the line passing through r₂ and having the direction n₂ at a point c, which corresponds to the center of the sphere. The intersection point is obtained by finding the point that is closest to both of the lines. In addition to this constraint, we have to ensure that

|∥c−r ₁ ∥−∥c−r ₂∥|<ε,  (9)

such that the points r₁ and r₂ are located at the same distance from the center of the sphere c. We use these constraints to remove false samples of the two oriented points: If the two lines do not intersect, or Equation (9) is not satisfied, then we discard the sampled points and restart the sampling process from the first point.

We determine the average distance from c to r₁ and r₂ as the radius of the sphere r, i.e.,

r=(∥c−r ₁ ∥+∥c−r ₂∥)/2  (10)

All points r that lie on the sphere should satisfy\

|∥c−r∥−r|<ε  (11)

The score for the sphere is determined by sampling several points on this sphere satisfying Equation (11) and averaging the distances of the sampled points using Equation (8).

Cylinder Primitives

For hypothesizing a cylinder, we first sample two oriented points (r₁,n₁) and (r₂,n₂) that define the axis and radius of the cylinder. The direction of the axis is defined as n_(a)=n₁×n₂, where × denotes the cross product. We project the two oriented points (r₁,n₁) and (r₂,n₂) along the direction of the axis n_(a) onto a common plane, and determine the intersection of two lines on the plane, defined by the two oriented points projected onto the plane. The two projected points should be approximately the same distance from the intersection point so that we can determine the radius of the cylinder using the average distance between the intersection and the projected two points. If this condition is not satisfied, then we discard the sampled points and restart the sampling process from the first point.

Next we sample other two oriented points (r₃,n₃) and (r₄,n₄) to define the two planar surfaces of the cylinder. The two points should satisfy a condition that n₃ and n₄ are parallel to the direction of the axis n_(a), i.e.,

1−|n _(a) ^(T) n ₃<ε and 1−|n _(a) ^(T) n ₄|<ε  (12)

If this condition is not satisfied, then we discard the sampled points and restart the sampling process from the first point.

To determine the score of the cylinder, we sample several points on the surface of the cylinder, and average the distances of the sampled points as in Equation (8).

Cuboid Primitives

A cuboid can be hypothesized by sampling six oriented points. We sample the first two points (r₁,n₁) and (r₂,n₂) such that the normals n₁ and n₂ are parallel to each other, i.e.,

1−|n ₁ ^(T) n ₂|<ε  (13)

We sample the next points (r₃,n₃) and (r₄,n₄) such that n₃ is perpendicular to n₁ and n₄ is parallel to n₃ i.e.,

|n ₁ ^(T) n ₃|<ε and 1−|n ₃ ^(T) n ₄|<ε  (14)

We sample the last two points such that (r₅,n₅) and (r₆,n₆) such that n₅ is perpendicular to both n₁ and n₃, and n₆ is parallel to n₅, i.e.,

|n ₁ ^(T) n ₅ |<ε, |n ₃ ^(T) n ₅|<ε, and 1−|n ₅ ^(T) n ₆|<ε  (15)

If any of the above conditions are not satisfied while sampling the six points, then we discard the sampled points and start the sampling process from the first point again.

Each of the six points defines a face of a cuboid. To determine the score, we sample a set of points on the six faces and average the distance values of the sampled points as in Equation (8).

Best Primitive Selection

For each iteration of our method, we hypothesize several primitive shapes and determine their corresponding scores as described above. Then, we select the best primitive that has the minimal score among all the hypotheses.

This naïve scoring method prefers a primitive shape with a small surface area. To avoid selecting such a small primitive shape, we use a score weighted by the surface area of the primitive shape for the best primitive selection. Specifically, we use v/S^(p) as the weighted score, where S denotes the surface area of the primitive, and p is a parameter to balance the weighting between the score and the surface area.

Refinement

We optionally refine 130 the parameters of the best primitive shape using a gradient-decent procedure. We start the gradient decent from the primitive parameters determined in the RANSAC process. For each gradient decent step, we sample several points r_(k) (k=1, . . . , k) on the surface of the primitive shape given the current primitive parameters, and use their gradient vectors ∇D(r_(k)) to determine a Jacobian matrix with respect to each parameter of the primitive shape for refining the parameters.

Subtraction

Given the best primitive, we subtract the points on the primitive from the distance field. We determine the distance between the surface of the primitive shape and each point in the set Z, and if the distance is within a small threshold, we remove the point from Z. We then determine the distance field D(x) by using the new set of points in Z and also update the gradient vector field. The new distance field and gradient vector field are used in the next iteration of our method.

Although the invention has been described by way of examples of preferred embodiments, it is to be understood that various other adaptations and modifications can be made within the spirit and scope of the invention. Therefore, it is the object of the appended claims to cover all such variations and modifications as come within the true spirit and scope of the invention. 

We claim:
 1. A method for fitting primitive shapes to a set of three-dimensional (3D) points, comprising the steps of: converting the set of 3D points to a distance field, wherein each element in the distance field is associated with a distance to a nearest point in the set of 3D points; hypothesizing a set of two or more candidates of the primitive shapes; determining a score for each candidate using the distance field; and selecting the candidate as the primitive shape to fit to the 3D points according to the scores, wherein the steps are performed in a processor.
 2. The method of claim 1, wherein the 3D points are acquired by a 3D sensor.
 3. The method of claim 1, further comprising: determining a gradient vector field of the distance field.
 4. The method of claim 3, wherein the hypothesizing uses a minimum number of the elements in the distance field required for determining parameters of each candidate.
 5. The method of claim 1, further comprising: fitting a set of primitive shapes by subtracting the selected candidate from the distance field and repeating the converting, hypothesizing, determining, and selecting.
 6. The method of claim 3, further comprising: refining the selected candidate by using a gradient-decent procedure, wherein the gradient-decent procedure uses the gradient vector field.
 7. The method of claim 1, wherein P={p_(i)}, for i=1, . . . , N is the set of 3D points, the distance field is D(x), where X represents a 3D voxel in a regular volume grid having the size of W×H×L, and R is a 3×3 rotation matrix, t is a 3×1 translation vector, and s is a scale factor to transform a coordinate system of the set of 3D points to a coordinate system of the distance field, and an i^(th) 3D point p_(i) is transformed to the coordinate system of the distance field as q_(i)=s(Rp_(i)+t), such that all q_(i) are within the distance field.
 8. The method of claim 7, wherein q_(i) are discretized to a nearest grid point of the distance field as q_(i) ^(d)=round(q_(i)), wherein the function round(·) determines the nearest grid point of the 3D point.
 9. The method of claim 8, wherein the distance field is determined by solving a minimization ${{D(x)} = {\min\limits_{y \in Q}\left( {{d\left( {x,y} \right)} + {T(y)}} \right)}},$ where a function d determines a distance between points x and y, and T is an indicator function ${T(y)} = \left\{ \begin{matrix} 0 & {{{if}\mspace{14mu} y} \in Q} \\ \infty & {{otherwise}.} \end{matrix} \right.$
 10. The method of claim 6, further comprising: determining a Jacobian matrix using the gradient vector field.
 11. The method of claim 3, wherein each element is an oriented 3D point having a 3D location and a 3D direction vector.
 12. The method of claim 1, wherein the primitive shapes are selected from a group consisting of infinite planes, plane segments, spheres, cylinders, cuboids, and combinations thereof.
 13. The method of claim 1, wherein the score is weighted by a surface area of the primitive shape. 